⭐ Sabse Important Points · एक नज़र में
- Ratio: A comparison of two quantities of the same unit, expressed as a:b, where b ≠ 0.
- Direct Proportion: Two quantities are directly proportional if their ratio is constant (y = kx).
- Proportion: An equality of two ratios, a:b = c:d, which means ad = bc (product of extremes equals product of means).
- Inverse Proportion: Two quantities are inversely proportional if their product is constant (xy = k).
- Simplification: Divide all terms by their greatest common divisor to get the simplest form.
- Partnership Ratio: Profit shares are proportional to the product of investment and time.
Introduction · परिचय
Ratio and Proportion is a fundamental topic in arithmetic that deals with comparing quantities and finding equivalent relationships. It is widely used in problems related to partnership, mixtures, ages, and data interpretation. For RRB NTPC, questions are typically direct and test your ability to simplify ratios, find missing terms, and apply proportional reasoning.
Basics of Ratio
A ratio compares two quantities of the same kind by division. It is expressed as a:b or a/b, where a and b are positive numbers. The ratio is always in simplest form, and the quantities must be in the same unit. A proportion is an equation stating that two ratios are equal, i.e., a:b = c:d, which implies a/b = c/d or ad = bc.
📐 Formula / Rule Box · सूत्र याद रखो
Proportion Rule
If a:b = c:d, then ad = bc
Continued Proportion
If a:b = b:c, then b² = ac
⭐ Key Points · याद रखने वाली बातें
- Ratio: A comparison of two quantities of the same unit, expressed as a:b, where b ≠ 0.
- Proportion: An equality of two ratios, a:b = c:d, which means ad = bc (product of extremes equals product of means).
- Simplification: Divide all terms by their greatest common divisor to get the simplest form.
- Unit Consistency: Quantities in a ratio must be in the same unit before comparison.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Simplify the ratio 24:36.
✅ Solution: 2:3
💡 Explanation: Divide both terms by their GCD, 12: 24÷12 = 2, 36÷12 = 3. So ratio is 2:3.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: If a:b = 3:4 and b:c = 5:6, find a:c.
✅ Solution: 5:8
💡 Explanation: Make b common: a:b = 15:20, b:c = 20:24, so a:b:c = 15:20:24. Thus a:c = 15:24 = 5:8.
Types of Proportion and Applications
Proportion can be direct or inverse. In direct proportion, as one quantity increases, the other increases at the same rate (e.g., more workers do more work). In inverse proportion, as one increases, the other decreases (e.g., more workers take less time). Ratio and proportion are applied in partnership (profit sharing), mixtures, and scale drawings.
| Direct Proportion | Inverse Proportion |
|---|---|
| Both increase or decrease together | One increases, other decreases |
| Ratio constant: y/x = k | Product constant: xy = k |
| Example: More workers, more work | Example: More workers, less time |
⭐ Key Points · याद रखने वाली बातें
- Direct Proportion: Two quantities are directly proportional if their ratio is constant (y = kx).
- Inverse Proportion: Two quantities are inversely proportional if their product is constant (xy = k).
- Partnership Ratio: Profit shares are proportional to the product of investment and time.
- Mixture Problems: Ratios help determine the composition of mixtures and alligation.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: If 8 workers can build a wall in 12 days, how many days will 6 workers take?
✅ Solution: 16 days
💡 Explanation: Workers and time are inversely proportional: 8×12 = 6×d ⇒ d = 96/6 = 16 days.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: A and B invest Rs. 5000 and Rs. 6000 respectively. After 1 year, profit is Rs. 2200. Find A's share.
✅ Solution: Rs. 1000
💡 Explanation: Ratio of investments = 5000:6000 = 5:6. Total parts = 11. A's share = (5/11)×2200 = 1000.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1If x:y = 2:3 and y:z = 4:5, find x:z.
👀 Answer dekho
✅ Answer: 8:15
💡 Explanation: Make y common: x:y = 8:12, y:z = 12:15, so x:y:z = 8:12:15. Thus x:z = 8:15.
Q2The ratio of two numbers is 5:7 and their sum is 72. Find the smaller number.
👀 Answer dekho
✅ Answer: 30
💡 Explanation: Let numbers be 5k and 7k. 5k+7k=12k=72 ⇒ k=6. Smaller number = 5×6=30.
Q3If 15 men can complete a work in 20 days, how many men are needed to complete it in 12 days?
👀 Answer dekho
✅ Answer: 25 men
💡 Explanation: Men and days are inversely proportional: 15×20 = m×12 ⇒ m = 300/12 = 25.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Forgetting to convert quantities to the same unit before forming a ratio (e.g., mixing metres and centimetres).
- In inverse proportion, mistakenly using direct proportion formula; always check if the quantities increase/decrease together or oppositely.
- In partnership, ignoring the time factor: profit share is proportional to investment × time, not just investment.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Ratio: Always express in simplest form and ensure same units.
- Proportion: a:b = c:d ⇒ ad = bc; use cross-multiplication to solve.
- Direct vs Inverse: Direct means both increase together; inverse means one increases while the other decreases.
- Partnership: Profit ratio = (Investment × Time) ratio.
- Continued Proportion: a:b = b:c ⇒ b² = ac.